Vibration Analysis of Non-uniform Beams Resting on Two Layer Elastic Foundations Under Axial and Transverse Load Using (GDQM)

The natural frequencies of non-uniform beams resting on two layer elastic foundations are numerically obtained using the Generalized Differential Quadrature (GDQ) method. The Differential Quadrature (DQ) method is a numerical approach effective for solving partial differential equations. A new combination of GDQM and Newton’s method is introduced to obtain the approximate solution of the governing differential equation. The GDQ procedure was used to convert the partial differential equations of non-uniform beam vibration problems into a discrete eigenvalues problem. We consider a homogeneous isotropic beam with various end conditions. The beam density, the beam inertia, the beam length, the linear (k1) and nonlinear (k2) Winkler (normal) parameters and the linear (k3) Pasternak (shear) foundation parameter are considered as parameters. The results for various types of boundary conditions were compared with the results obtained by exact solution in case of uniform beam supported on elastic support.


Introduction
Beams resting on linear and non-linear elastic foundations have many practical engineering applications as railroad tracks, highway pavement, buried pipelines and foundation beams. Due to the difficulty of mathematical nature of the problem, a few analytical solutions limited to special cases for vibrations of non-uniform beams resting on non-linear elastic foundations are found. Many methods are used to obtain the vibration behavior of different types of linear or nonlinear beams resting on linear or nonlinear foundations such as finite element method [1][2][3], transfer matrix method [4], Rayleigh-Ritz method [5], differential quadrature element method (DQEM) [6][7][8][9][10], Galerkin procedure [11,12] and [13][14][15]. There are various types of foundation models such as Winkler, Pasternak, Vlasov, etc. that have been used in the analysis of structures on linear and non-linear elastic foundations. Also, There are different beam types such as the Euler-Bernoulli which for slender beams and Timoshenko beam model for moderately short and thick beams. Balkaya et al. [16] studied vibration of a uniform Euler beam on elastic foundation using Differential Transform Method. Also, Ozturk and Coskun [17] studied the same problem using HPM. Avramidis and Morfidis [18] analyzed bending of beams on three-parameter elastic foundation.
Abrate et al. [19,21] studied the vibrations of non-uniform rods and beams using the Rayleigh-Ritz scheme. Hodges et al. [22] used a discrete transfer matrix scheme to compute the fundamental frequencies and the corresponding modal Under Axial and Transverse Load Using (GDQM) shapes. Sharma and DasGupta [23] used the Green's functions to study the bending of axially constrained beams resting on nonlinear Winkler type elastic foundations. Beaufait and Hoadley [24] used the midpoint difference technique to solve the problem of elastic beams resting on a linear foundation. Kuo and Lee [25] used the perturbation method to investigate the deflection of non-uniform beams resting on a nonlinear elastic foundation. Chen [26] used the differential quadrature element approach to obtain the numerical solutions for beams resting on elastic foundations.
Bagheri et al. [27] studied the nonlinear responses of clamped-clamped buckled beam. They used two efficient mathematical techniques called variational approach and Laplace iteration method in order to obtain the responses of the beam vibrations. Nikkar et al. [28] studied the nonlinear vibration of Euler-Bernoulli using analytical approximate techniques. Li and Zhang [29] used the B-spline function to derive a dynamic model of a tapered beam. Ramzy et al. [30] presented a new technique of GDQM for determining the deflection of a non-uniform beam resting on a non-linear elastic foundation, subjected to axial and transverse distributed force. Ramzy et al. [31] presented some problems in structural analysis resting on fluid layer using GDQM. Ramzy et al. [32] studied free vibration of uniform and non-uniform beams resting on fluid layer under axial force using the GDQM. The details of the DQM and its applications can be found in [33][34].
From previous studies, there are no any attempts to study vibration of non-uniform beam resting on two non-linear elastic foundations with the linear and nonlinear Winkler (normal) parameters and the linear Pasternak (shear) foundation parameter. The main goal of this study, to present a new combination of a GDQM and Newton's method to obtain the fundamental frequencies and the corresponding modal shapes of non-uniform beams resting on two layer elastic foundations under appropriate boundary conditions.

Formation
The kinetic energy of a beam with a non-uniform crosssection resting on an elastic foundation is as follows: Where the length of the non-uniform beam L, the vertical displacement v, the cross-sectional area of the beam A and the density of the beam material ρ.
The strain energy of a non-uniform beam resting on an elastic foundation can be derived as follows: Where the inertia of the beam I, the constant of the foundation k and Young's modulus of the beam material E.
Due to the axial loading, the work done can be written as follows, Where, p is the axial force. The Hamilton's principle is given by: where δW is the virtual work. Substituting from Equations (1), (2) and (3) into Equation (4) yields, Equation (5) represents the equation of motion of nonuniform beam resting on elastic foundations under axial force.
The corresponding boundary conditions are as follows: For Clamped-Clamped supported (C-C); For Simply-Simply supported (S-S); For Clamped-Simply supported (C-S); The vibration equation of a flexural non-uniform beam resting on two-layer elastic foundation is given as: To obtain the natural frequencies and mode shapes, one can assume: Where the amplitude of free vibration V(x), the natural frequency of the beam ω and the external dynamic distributed load applied q(x, t). Substituting form Equations (13) and (14) into Equation (12) yields Through the normalization process we can transform Equation (16) into non-dimensional form as follows, The non-dimensional coefficients are: Where the non-dimensional deflection of the beam W, the non-dimensional axial loading P, the non-dimensional foundation linear stiffness K, the non-dimensional frequency of the beam Ω, the beam's flexural rigidity EI, the mass per unit length ρA and the inertia ratio S(X).
Equation (17) is a 4 th order ordinary differential equation with inertia ratio In this section, we will study two cases of inertia ratio S(X); the first case α 1 = 0.5, α 2 =1 and the second case α 1 = −1, α 2 =1.

Solution of the Problem
The method of GDQ is employed to solve the problem. This method requires to descretize the domain of the problem into N pointes. Then the derivatives at any points are approximated by a weighted linear summation of all the functional values along the descretized domain, as follows [33][34]: Where, A ij represented the weighting coefficient, and N is the number of grid points in the whole domain. Equation (18) is called Differential Quadrature (DQ) technique. It should be noted that the weighting coefficients A ij are different at different location of x i . The key to DQ is to determine the weighting coefficients for the discretization of a derivative of any order.
The weighting coefficient can be determined by making use of Lagrange interpolation formula as follows: By applying Equation (19) at N grid points, they obtained the following algebraic formulations to compute the weighting coefficients A ij . 1, , For calculating the weighting coefficients of m th order . .
The accuracy of the results obtained by DQM, is affected by choosing of the number of grid points, N, and the type of sampling points, x i . It is found that the optimal selection of the sampling grid points in the vibration problems, are chosen according to Gauss-Chebyshev-lobatto points [33][34], Applying the GDQ discretization scheme to the nondimensional governing Equation (17) yields; (1) (2) Where i W is the functional value at the grid points i X .
ij B , ij C and ij D is the weighting coefficient matrix of the second, third and forth order derivatives.
Assuming that the external dynamic distributed load changes as the deflection amplitude change, then the governing equations system (39) can be written as It is noted that Equations (40) has (N−4) equations with (N−4) unknowns, which can be written in matrix eigen-value form as

Results
In this section forced vibration of Euler-Bernoulli of nonuniform beams resting on two layer elastic foundations under axial and transverse load is analyzed. The GDQM is used to compute the first three natural frequencies and the corresponding mode shapes for the forced vibration of nonuniform under axial and transverse force with two cases of inertia ratio 2 1 ( ) (1 ) S X X α α = + , the first case: 1 α 1 = 0.5, α 2 = 1.0 and the second case: α 1 = -1.0, α 2 = 1.0, with three different types of end conditions. Fifteen non-uniformly spaced grid points were chosen by the previous relation.

Accuracy and Stability
In order to discuss the stability and accuracy of the GDQM, uniform beams are solved using the present approach for implementing the boundary conditions and the results are compared with the exact results available in the literature. The exact solutions are introduced as found in Qiang [34] and Blevins [35] for uniform beam S(X) = 1.0. The results are presented in Tables (1) through (3).
Tables (1) through (3) show the first three non-dimensional natural frequencies of uniform beam Clamped-Clamped Beam (C-C) Supported, Simply-Simply (S-S) Beam Supported and Clamped-Simply Beam (C-S) supported, respectively. Fifteen non-uniformly spaced grid points were chosen by the previous relation.
The absolute relative error typed in Tables (1) through (3) represents the accuracy of the GDQM. This absolute relative error can be defined by the formula, Present-Exact 100 Examining the three tables.

Results Using a Proposed Technique of GDQM
Tables (4) and (5) show the first three non-dimensional natural frequencies of non-uniform beam resting on two layer elastic foundations with two cases of inertia ratio It can be observed from Table (4) and (5) that, the natural frequencies increase when the beam resting on two layer Under Axial and Transverse Load Using (GDQM) elastic foundations.
The corresponding mode shapes are presented in Figures  (2-7). Figures (2-4) for the first case of inertia ratio and Figures (5-7) for the second case of inertia ratio.
Case (1): Case (2):      To examine the effect of the non-linear elastic foundation "K 2 ", we fix the other values of elastic foundations "K 1 " and "K 3 ", the value of axial load "P" and the value of distributed dynamic force "F". Then draw "K 2 " versus the natural frequency. It is clear that increasing the non-linear elastic foundation "K 2 " increases the natural frequency of the beam, Figures (8-10) for the first case of inertia ratio and Figures  (11-13) for the second case of inertia ratio.

Conclusion
In this paper, an efficient algorithm based on a new combination of a GDQM and Newton's method presented for solving eigenvalue problems of non-uniform beams resting on two layer elastic foundations. Appropriate boundary conditions and the GDQM are applied to transform the partial differential equations of non-uniform beams resting on two layer elastic foundations into discrete eigenvalue problems. The results for various types of boundary conditions were compared with the results obtained by exact solution in case of uniform beam supported on elastic support.
From the parametric study of nonlinear elastic foundation of vibration analysis for various types of boundary conditions, that,